‎In this paper we first obtain the non-Riemannian Randers metrics of Douglas type on two-step homogeneous nilmanifolds of dimension five‎. ‎Then we explicitly give the flag curvature formulae and the $S$-curvature formulae for the Randers metrics of Douglas type on these spaces‎. ‎Moreover‎, ‎we prove that the only simply connected five-dimensional two-step homogeneous Randers nilmanifolds of D...

In this paper we investigate expanding maps on infra-nilmanifolds. Such manifolds are obtained as a quotient E\L, where L is a connected and simply connected nilpotent Lie group and E is a torsion-free uniform discrete subgroup of LoC, with C a compact subgroup of Aut(L). We show that if the Lie algebra of L is homogeneous (i.e., graded and generated by elements of degree 1), then the correspon...

Two Riemannian manifolds are said to have C-conjugate geodesic flows if there exist an C diffeomorphism between their unit tangent bundles which intertwines the geodesic flows. We obtain a number of rigidity results for the geodesic flows on compact 2-step Riemannian nilmanifolds: For generic 2-step nilmanifolds the geodesic flow is C rigid. For special classes of 2-step nilmanifolds, we show t...

We give a necessary and sufficient condition for k-step nilmanifolds associated with graphs (k ≥ 3) to admit Anosov automorphisms. We also prove nonexistence of Anosov automorphisms on certain classes of 2-step and 3-step nilmanifolds. 2000 Mathematics Subject Classification. Primary 37D20; Secondary 22E25